Tagged Questions
4
votes
2answers
108 views
geometric interpretation of Lie bracket
On page 159 of "A Comprehensive Introduction To Differential Geometry Vol.1" by Spivak has written:
We thus see that the bracket $[X,Y]$ measures, in some sense, the extent to
which the integral ...
3
votes
1answer
87 views
Exponential map on manifolds and differential
I am trying to understand the proof of Theorem 3.7, page 72 of Riemannian Geometry by M. Do Carmo.
For $M$ a Riemannian manifold and $(U,\varphi)$ a chart around a point
$p\in M$, he (more or ...
1
vote
1answer
132 views
Commutators, and Christoffel symbols in a non holonomic basis
I have a frame that varies along a curve $\gamma$ : the frame consists in the tangent vector of the curve plus some constant non orthogonal vectors.
I need to compute ...
3
votes
1answer
131 views
Derivative of a parallel translation inside a metric
Let $M$ be a riemannian manifold with metric $g$ and a connection
$\nabla$ on $M$. Let $X,Y$ two vector fields along a curve $\gamma$
on $M$. Let $$\tau_{t,s}:T_{\gamma(s)}M\to ...
2
votes
1answer
84 views
Derivative of a metric tensor along a curve
Let $M$ be a Riemannian manifold with metric tensor $g$ and Levi-Civita connection $\nabla$.
Also, let $u: \mathbb{R}\to T_pM$ be a smooth curve in $T_pM$. In a proof, my course notes assure that
...
